{
  "title": "Directional VOA, module, finiteness, smearing, net, and comparison theorems",
  "scope": "Qualitative theorem and construction comparison; no quantitative data are tabulated.",
  "table_id": "vertex-operator-algebras-net-bridges-hypothesis-conclusion-table",
  "caption": "Directional VOA, module, finiteness, smearing, net, and comparison theorems",
  "columns": [
    {
      "text": "Object and domain",
      "html": "Object and domain"
    },
    {
      "text": "Required hypotheses",
      "html": "Required hypotheses"
    },
    {
      "text": "Licensed conclusion",
      "html": "Licensed conclusion"
    },
    {
      "text": "Excluded converse or upgrade",
      "html": "Excluded converse or upgrade"
    },
    {
      "text": "Adversarial check",
      "html": "Adversarial check"
    }
  ],
  "rows": [
    {
      "Object and domain": {
        "text": "Graded state–field correspondence",
        "html": "Graded state–field correspondence"
      },
      "Required hypotheses": {
        "text": "Vacuum and creation, translation, lower truncation, conformal grading, Jacobi or uniform formal locality",
        "html": "Vacuum and creation, translation, lower truncation, conformal grading, Jacobi or uniform formal locality"
      },
      "Licensed conclusion": {
        "text": "A vertex operator algebra with all mode identities",
        "html": "A vertex operator algebra with all mode identities"
      },
      "Excluded converse or upgrade": {
        "text": "A finite mode table or several OPE coefficients do not establish the Jacobi identity",
        "html": "A finite mode table or several OPE coefficients do not establish the Jacobi identity"
      },
      "Adversarial check": {
        "text": "Alter a high mode while preserving every tested low mode and look for a missing locality exponent",
        "html": "Alter a high mode while preserving every tested low mode and look for a missing locality exponent"
      }
    },
    {
      "Object and domain": {
        "text": "Modules and intertwining operators",
        "html": "Modules and intertwining operators"
      },
      "Required hypotheses": {
        "text": "Closure under tensor products, convergence and extension of products and iterates, parallel transport, grading restrictions",
        "html": "Closure under tensor products, convergence and extension of products and iterates, parallel transport, grading restrictions"
      },
      "Licensed conclusion": {
        "text": "Associativity, braiding, and a coherent tensor category in the stated module class",
        "html": "Associativity, braiding, and a coherent tensor category in the stated module class"
      },
      "Excluded converse or upgrade": {
        "text": "Fusion multiplicities alone do not determine associators, braiding, rigidity, or positivity",
        "html": "Fusion multiplicities alone do not determine associators, braiding, rigidity, or positivity"
      },
      "Adversarial check": {
        "text": "Supply an associative based ring but no analytic continuation of insertion points",
        "html": "Supply an associative based ring but no analytic continuation of insertion points"
      }
    },
    {
      "Object and domain": {
        "text": "Finiteness and semisimplicity",
        "html": "Finiteness and semisimplicity"
      },
      "Required hypotheses": {
        "text": "Declared CFT-type, $C_2$-cofiniteness, rationality, self-contragredience, and any rigidity hypotheses",
        "html": "Declared CFT-type, $C_2$-cofiniteness, rationality, self-contragredience, and any rigidity hypotheses"
      },
      "Licensed conclusion": {
        "text": "Regularity or a finite modular tensor category at exactly the theorem's strength",
        "html": "Regularity or a finite modular tensor category at exactly the theorem's strength"
      },
      "Excluded converse or upgrade": {
        "text": "$C_2$-cofiniteness alone does not imply rationality or regularity",
        "html": "$C_2$-cofiniteness alone does not imply rationality or regularity"
      },
      "Adversarial check": {
        "text": "Use a triplet VOA with logarithmic indecomposable modules",
        "html": "Use a triplet VOA with logarithmic indecomposable modules"
      }
    },
    {
      "Object and domain": {
        "text": "Zhu algebra and genus-one traces",
        "html": "Zhu algebra and genus-one traces"
      },
      "Required hypotheses": {
        "text": "Admissible top levels; rationality and $C_2$ control for ordinary modular traces, or generalized pseudotrace hypotheses",
        "html": "Admissible top levels; rationality and $C_2$ control for ordinary modular traces, or generalized pseudotrace hypotheses"
      },
      "Licensed conclusion": {
        "text": "Simple top-level classification and finite-dimensional modular covariance of the appropriate trace space",
        "html": "Simple top-level classification and finite-dimensional modular covariance of the appropriate trace space"
      },
      "Excluded converse or upgrade": {
        "text": "A finite Zhu algebra need not be semisimple, and simple characters need not close in a logarithmic theory",
        "html": "A finite Zhu algebra need not be semisimple, and simple characters need not close in a logarithmic theory"
      },
      "Adversarial check": {
        "text": "Retain a nilpotent $L_0$ part and compare characters with pseudotraces",
        "html": "Retain a nilpotent $L_0$ part and compare characters with pseudotraces"
      }
    },
    {
      "Object and domain": {
        "text": "Unitary VOA on its Hilbert completion",
        "html": "Unitary VOA on its Hilbert completion"
      },
      "Required hypotheses": {
        "text": "Positive invariant form, PCT adjoints, polynomial mode bounds, common invariant smooth core",
        "html": "Positive invariant form, PCT adjoints, polynomial mode bounds, common invariant smooth core"
      },
      "Licensed conclusion": {
        "text": "Closable smeared vertex operators with controlled adjoints and energy domains",
        "html": "Closable smeared vertex operators with controlled adjoints and energy domains"
      },
      "Excluded converse or upgrade": {
        "text": "Formal locality or a vanishing core commutator does not imply strong commutativity of closures",
        "html": "Formal locality or a vanishing core commutator does not imply strong commutativity of closures"
      },
      "Adversarial check": {
        "text": "Choose symmetric operators commuting on one dense core but with noncommuting spectral resolutions",
        "html": "Choose symmetric operators commuting on one dense core but with noncommuting spectral resolutions"
      }
    },
    {
      "Object and domain": {
        "text": "Strongly local simple unitary VOA",
        "html": "Strongly local simple unitary VOA"
      },
      "Required hypotheses": {
        "text": "Energy bounds and strong locality for every interval, plus the vacuum and conformal-vector hypotheses",
        "html": "Energy bounds and strong locality for every interval, plus the vacuum and conformal-vector hypotheses"
      },
      "Licensed conclusion": {
        "text": "An irreducible diffeomorphism-covariant conformal net and recovery within the theorem's image",
        "html": "An irreducible diffeomorphism-covariant conformal net and recovery within the theorem's image"
      },
      "Excluded converse or upgrade": {
        "text": "Not every unitary VOA is known to be strongly local, and the theorem is not a universal net–VOA equivalence",
        "html": "Not every unitary VOA is known to be strongly local, and the theorem is not a universal net–VOA equivalence"
      },
      "Adversarial check": {
        "text": "Remove energy bounds and show that the smeared series lacks a controlled closure",
        "html": "Remove energy bounds and show that the smeared series lacks a controlled closure"
      }
    },
    {
      "Object and domain": {
        "text": "Extensions, orbifolds, cosets, and converse maps",
        "html": "Extensions, orbifolds, cosets, and converse maps"
      },
      "Required hypotheses": {
        "text": "Compatible algebra objects or subobjects, positivity and locality, integrability, finite-index or Condition-I/II hypotheses, matched equivalence notions",
        "html": "Compatible algebra objects or subobjects, positivity and locality, integrability, finite-index or Condition-I/II hypotheses, matched equivalence notions"
      },
      "Licensed conclusion": {
        "text": "The specified extension comparison, sector functor, or partial net-to-VOA reconstruction",
        "html": "The specified extension comparison, sector functor, or partial net-to-VOA reconstruction"
      },
      "Excluded converse or upgrade": {
        "text": "Matching modular data, characters, or sector branching do not establish equality of extensions or essential surjectivity",
        "html": "Matching modular data, characters, or sector branching do not establish equality of extensions or essential surjectivity"
      },
      "Adversarial check": {
        "text": "Keep the branching matrix but remove the multiplication, energy bounds, or point-field generation theorem",
        "html": "Keep the branching matrix but remove the multiplication, energy bounds, or point-field generation theorem"
      }
    }
  ]
}